Re: Algebraic Theories/Operads

Re: Algebraic Theories/Operads

[categories]

Date: Mon, 8 Dec 1997 20:15:23 +1100 (EST)
From: Barry Jay <cbj@socs.uts.EDU.AU>


Tom,

you may care to look at my paper "Languages for monoidal categories"

@Article(Jay89b,
	Author={Jay, C.B.},
	Title={Languages for monoidal categories},
	Journal=jpaa,
	Volume=59,
	Year=1989,
	Pages={61--85})

and its successors (see my ftp site below).

Barry Jay


*************************************************************************
| Associate Professor C.Barry Jay, 					|
| Reader in Computing Sciences		Phone: (61 2) 9514 1814		|
| Head, Algorithms and Languages Group,	Fax:   (61 2) 9514 1807		|
| University of Technology, Sydney,	e-mail: cbj@socs.uts.edu.au	|
| P.O. Box 123 Broadway, 2007,		www: linus.socs.uts.edu.au/~cbj	|
| Australia.			     ftp: ftp.socs.uts.edu.au/Users/cbj |
*************************************************************************

Re: Algebraic Theories/Operads

[categories]

Date: Fri, 5 Dec 1997 18:54:36 +0000 (GMT)
From: Tom Leinster <T.Leinster@dpmms.cam.ac.uk>

>
> I'm looking for a good reference on the relation between
> algebraic theories (a la Lawvere) and operads.
>
> David Metzler

I think the relationship between algebraic theories and (non-symmetric,
non-topological) operads is quite simply described. Loosely, algebras for
operads are just the same as algebras for strongly regular theories. To be
more precise: any operad gives rise to a monad on Set, the algebras for which
are the algebras for the operad; a monad on Set arises from an operad iff it
arises from a strongly regular theory. So -

	operadic monads are strongly regular theories.


Carboni & Johnstone (1995) call an equation _strongly regular_ if on each
side the same variables appear in the same order, without repetition (e.g.
(x.y).z=x.(y.z) and x-(y-z)=(x-y)+z, but not x.y=x, x.y=y.x or
(x.x).y=x.(x.y)). A theory is called strongly regular if it can be presented
by operators and strongly regular equations - for instance, the theory of
monoids. They show that a monad (T, eta, mu) on Set is from a s.r. theory
("is s.r.") iff
	(i)   T is finitary
	(ii)  T preserves wide pullbacks
      & (iii) eta and mu are cartesian.
(A wide pullback is a limit over a diagram (X_i --> X) where i ranges over
some set I, e.g. if I=2 it's an ordinary pullback.)

1. Operadic => strongly regular

If A is an operad, with the function A --> N={natural numbers}, then the
functor part of the induced monad T on Set is defined by the pullback square

	T(X) ----> W(X)
	 |          |
	 |          | W(!)
	 |          |
	 V          V
	 A ------> W(1)=N

 where X is a set and W (for Words) is the free-monoid monad. One can show
(e.g. my (1997, sec 4.6)) that the unit and multiplication of T are
cartesian. Moreover, one can also show that
	(a) if W preserves colimits of a given shape then so does T
      & (b) if W preserves I-ary pullbacks then so does T.
Since the theory of monoids is s.r., W preserves all filtered colimits (i.e.
is finitary) and all wide pullbacks. So T satisfies (i)-(iii) and is
therefore s.r.. 

2. Strongly regular => operadic

Conversely, take a s.r. theory T. Any s.r. presentation of T gives rise to a
natural transformation T --> W which is cartesian and preserves the monad
structure. It follows by my (1997, sec 4.6) that the monad T comes from some
operad A.


References:

A Carboni, P T Johnstone (1995), Connected limits, familial representability 
        and Artin glueing. Math Struct in Comp Science, vol 5, 
        pp 441-459.
T Leinster (1997, updated 3 Dec), General operads and multicategories.
        http://www.dpmms.cam.ac.uk/~leinster.


I've heard tell that these ideas were explored by Kelly in his work on
clubs - can anyone enlighten me?


Tom Leinster

RE: Algebraic Theories/Operads

[categories]

Date: Thu, 27 Nov 1997 16:10:31 MET
From: Heinrich.Kleisli@unifr.ch

May, J. P.: Operads, algebras and modules. Contemp. Math. 202, 15-31 (1997).

Re: Algebraic Theories/Operads

[categories]

Date: Thu, 27 Nov 1997 10:40 +0530
From: CAYLEY@tifrvax.tifr.res.in


	The volume 202 of Contemprory mathematics series
	of the AMS titled

	Operads: Proc. of Renaissance conferences

	ed by J.Loday et al is a possible source.

				P.S.Subramanian,
				Tata Institute.

Re: Algebraic Theories/Operads

[categories]

Date: Wed, 26 Nov 1997 16:22:14 -0800 (PST)
From: john baez <baez@math.ucr.edu>

David Metzler writes:
 
> I'm looking for a good reference on the relation between
> algebraic theories (a la Lawvere) and operads.

Me too!  The closest thing I know is Boardman and Vogt's
book on homotopy invariant algebraic structures --- which
uses the framework of PROPs rather than operads.

Algebraic Theories/Operads

[categories]

Date: Wed, 26 Nov 1997 08:57:36 -0600 (CST)
From: David Metzler <metzler@math.rice.edu>

I'm looking for a good reference on the relation between
algebraic theories (a la Lawvere) and operads.

David Metzler